Eduardo Manuel Alvarez Observatorio los Algarrobos, Salto, Uruguay

1. Diurnal Parallax Determinations of Asteroid DistancesUsing Only Backyard Observations from a Single Station Eduardo Manuel Alvarez Observatorio los Algarrobos, Salto, Uruguay Robert Buchheim Altimira Observatory, Coto de Caza, CA USA SAS XXXI Symposium – Big Bear Lake, CA, USA – May 23-24, 2012

2. Subject • New method for measuring asteroid parallax from a single site • simple, accurate, self-contained • three mathematical insights • Technical base supporting the new method • Some practical examples

3. Single-site “Diurnal Parallax” Earth’s rotation provides a “baseline” so that a single observer can measure the parallax. Parallax angle φ(t) = [ RAtopocentric – RAgeocentric ] cosδ

4. Objectives: Demonstrate measurement of distance to asteroid single-site diurnal parallax “Self-Contained” method All necessary data observable from single site no need for ephemeris or external data Challenges: Infer “geocentric” position from “topocentric” observations. Asteroids move … rapidly! “back-out” secular motion, leaving only parallax motion Accuracy vs. Simplification: measurements & calculations Objectives & Challenges

5. μ ν Z= R∙sinδ D R observer δ z R∙cosδ y α x Geometry of Model • Geocentric RA = Topocentric RA when target is at transit • use sequential transits to measure geocentric RA rate (ν) • + Assume: • Geocentric distance ≈ constant • geocentric Dec rate ≈ topocentric Dec rate • Secular Geocentric rates are (approx.) constant

6. RAgeo rate estimated by RA(t2) – RA(t1) ν≈ ------------------- t2 – t1 Estimate Geocentric RA ratefrom Consecutive Transit Observations night #2 “RA rate ν” (measured) night #1 RA position time

7. Found parallax = 7.52 arc-sec Found distance = 0.997 AU True distance = 0.973 AU error = 2.5% Pretty good! Example: 8106 Carpino parallax, arc-sec elapsed time, hr

8. Estimating Geocentric RA rateUsing Consecutive Transit Observations… but not at opposition night #2 night #1 RA position “true” geocentric RA motion (not measurable) time

9. Constant-RA-rate linear approximation of “true” RA(t) curve night #2 culmination “RA rate ν” (measured) night #1 culmination RA position “true” geocentric RA motion (not measurable) time

10. On night #1, νover-estimates the RA rate … Compensating errors night #2 culmination night #1 culmination RA position time

11. On night #1, ν over-estimates the RA rate … On night #2, νunder-estimates the RA rate Compensating errors night #2 culmination night #1 culmination RA position time

12. Wonderfully, Using both nights, the errors in calculated parallax / distance tend to cancel out, resulting in a very accurate (average) distance estimate. Compensating errors night #2 culmination night #1 culmination RA position time

13. Found parallax = 2.8 arc-sec Found distance = 2.67 AU True distance = 2.545 AU Error = 5.1% Distant target: 414 Liriope parallax, arc-sec elapsed time, hr

14. Found parallax = 145 arc-sec Found distance = 0.052 AU True distance = 0.049 AU Error = 6.1% Fast-mover, Near-Earth Asteroid (162421) 2000 ET70 parallax, arc-sec elapsed time, hr

15. Each night, many data points to fit sine-curve Two nights, two short sets of observations each night: early in night at transit How many data points do you really need? “Four-Point Shortcut”

16. Each night, many data points to fit sine-curve Two nights, two short sets of observations each night: early in night at transit How many data points do you really need? With good astrometric images, “four-point shortcut” gives excellent distance estimates “Four-Point Shortcut”

17. Conclusions:Measurement of Asteroid Parallax • A feasible “backyard” project • Modest equipment, standard software • Replicate historically-important observations and calculations: • observe diurnal parallax effect • measure distance to solar-system object • determine scale of Solar System (“Astronomical Unit”) • Mathematical insights simplify: • Observations early in evening, and at transit • then go to bed … • Two consecutive nights provide sufficient data

18. Questions?